Problem 69 - Algebra and Trigonometry

Question

Describe how to locate the foci of the graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

Step-by-Step Solution

Answer

The foci of the hyperbola are located at the points \((-\sqrt{10}, 0)\) and \((\sqrt{10}, 0)\).

Step 1: Identify the Values for a and b

From the given equation, the values for a and b can be identified by taking the square root of the denominators for the x and y terms respectively. Here, \(a^{2} = 9\) and \(b^{2} = 1\) which gives \(a = 3\) and \(b = 1\).

Step 2: Calculate the Value for c

We use the formula for a hyperbola to find c: \(c = \sqrt{a^{2} + b^{2}}\). Plug in the values for a and b to get: \(c = \sqrt{3^{2} + 1^{2}} = \sqrt{10}\).

Step 3: Locate the Foci

The foci of the hyperbola are located at points (±c, 0) from the center (0, 0). Given that \(c = \sqrt{10}\), the foci will be located at the points \((-\sqrt{10}, 0)\) and \((\sqrt{10}, 0)\).